K-theory of the C∗-algebra of multivariable Wiener-Hopf operators associated with some polyhedral cones in Rn

Abstract

We consider th $C^{\ast }$-algebra $WH(R^n,P)$ of the multivariable Wiener-Hopf operators associated with a polyhedral cone in $R^n$ and the extension $0\to \mathcal{K}\to WH(R^n,P)\to WH(R^n,P)/\mathcal{K}\to 0$. The main theorem states that if $P$ satisfies suitable geometric conditions (satisfied, e.g., for all simplicial cones and the cones in $R^n,n\le 3$), then $K_{\ast }(WH(R^n,P))=(0,0);K_{\ast }(WH(R^n,P)/\mathcal{K})=(0,Z)$ and that the index map is an isomorphism. In the cource of the proof we construct a Fredholm operator in $WH(R^n,P)$ with an index 1. The proof is inductive and uses the Mayer-Vietoris exact sequence and the standart six term exact sequence in $K$-theory.